Thus, we have found that the eigenvalue spectrum consists of a large peak within the range \([\lambda _-,\lambda _+]\)  and a number of extra zero eigenvalues when \(\chi _A>\chi _B\) . This is the famous Marchenko-Pastur (MP) distribution. It describes the singular value distribution of a random \(m\times n\)  matrix when both \(m,n\gg 1\) . This result was first shown by Page [1]}. Using the spectrum, the entanglement entropy is given by
\(S(A) = -\operatorname{tr}\rho _{A} \ln \rho _A ={\left\lbrace \begin{array}{ll}\log (\chi _A) - \frac{\chi _A}{2\chi _B}, \quad &\chi _A < \chi _B \\\log (\chi _B) - \frac{\chi _B}{2\chi _A}, \quad &\chi _B < \chi _A. \\\end{array}\right.}\) 
